Good data structure for hyperbolic tiling

Question

Say you're doing something computational where each data point is a tile in a (not necessarily Euclidean) 2-dimensional tiling, for instance, a Life-like cellular automata. You might want a data structure where tiles that are near in distance are also near each other in memory. You might choose your tiling based on what's easy to represent this way.

For a tiling of the Euclidean plane, you probably have square tiles backed by a 2d array with array point a[i][j] = tile $a_{ij}$; you might have hex tiles instead, which can also be backed by a 2d array as explained in http://www.redblobgames.com/grids/hexagons/.

For a tiling of the sphere, it's pretty common to start with a sub-triangulation of the icosahedron and subdivide it into 5 2d arrays plus 2 points for the north pole, as explained in http://kiwi.atmos.colostate.edu/BUGS/geodesic/text.html. It looks like a geodesic dome.

Is there a 2d hyperbolic tiling that admits a representation in a nice data structure? The only software I've ever seen that uses hyperbolic tiling in this way is the game HyperRogue (a roguelike on the hyperbolic plane). It just has an object for each tile, and each tile keeps a record of its neighbors. It works but it's not pretty.

Answer 1

I've been dealing with hyperbolic tilings and ornaments in my Ph.D. thesis, soon to be published. There I concentrated on triangle reflection groups, i.e. tilings of triangles where each triangle edge is a symmetry axis. But the general concept can be applied to different tilings as well, and furthermore all tilings by regular polygons can be seen as subgroups of these triangle reflection groups.

My approach was the following: choose a central triangle, and label the reflections in its edges as $a,b,c$. Then every triangle in the tiling is the result of that central triangle after some sequence of these operations has been applied. So every string consisting of these three letters, possibly with repetitions, describes a single triangle. Conversely, for every triangle there will be several strings (corresponding to different paths from the center to the given triangle), but these strings can be ordered (e.g. using shortlex ordering) so you can choose the minimal one as the canonical representative of that triangle.

Starting from a formulation of the above as a Coxeter group, one can easily see a term rewrite system in these generators which identifies common paths. One can then use the Knuth-Bendix procedure to complete that system, and Aho-Corasick matching to turn that into an automaton which can be used to identify rewrite positions, or (if you omit all terminal states) to enumerate all triangles in a breadth-first search.

So the data structure I'd propose is this: implement a map from such strings to triangles. If the map keeps items which are closely related in shortlex order newar one another in memory, then you'd have some degree of memory locality as well.

Note: your question appears to be related to Symbolic coordinates for a hyperbolic grid?